Ke/Pe Homework #2

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Learning Objectives

After this lesson, students should be able to:

  • Recognize that engineers need to understand the many different forms of energy in order to design useful products
  • Explain the concepts of kinetic and potential energy.
  • Understand that energy can change from one form into another.
  • Understand that energy can be described by equations.

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Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

NGSS: Next Generation Science Standards - Science
  • Develop a model to describe that when the arrangement of objects interacting at a distance changes, different amounts of potential energy are stored in the system. (Grades 6 - 8) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object. (Grades 6 - 8) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
Common Core State Standards - Math
  • Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (Grade 6) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (Grades 9 - 12) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (Grades 9 - 12) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
International Technology and Engineering Educators Association - Technology
Colorado - Math
  • Solve real-world and mathematical problems involving the four operations with rational numbers. (Grade 7) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Reason quantitatively and use units to solve problems. (Grades 9 - 12) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Use units as a way to understand problems and to guide the solution of multi-step problems. (Grades 9 - 12) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
  • Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12) Details...View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
Colorado - Science
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Introduction/Motivation

Begin by showing the class three items: 1) an item of food (such as a bagel, banana or can of soda water), 2) a battery, and 3) you, standing on a stool or chair. Ask the class what these three things have in common. The answer is energy. The food contains chemical energy that is used by the body as fuel. The battery contains electrical energy (in the form of electrical, potential or stored energy), which can be used by a flashlight or a portable CD player. A person standing on a stool has potential energy (sometimes called gravitational potential energy) that could be used to crush a can, smash the banana, or really hurt the foot of someone standing under you. Do a dramatic demonstration of jumping down on the banana or an empty soda can. (Be careful! Banana peels are slippery!) Explain the ideas of potential energy and kinetic energy as two different kinds of mechanical energy. Give definitions of each and present the equations, carefully explaining each variable, as discussed in the next section,

PE = mass x g x height

and

Explain how energy can be converted from one form to another. This should be clear from the jumping demonstration. You had potential energy (stored energy) when standing on the stool, which completely changed into kinetic energy (energy of motion) right before you landed on the ground. As a side note, the ground absorbed your energy when you landed and turned it into heat.

Lesson Background and Concepts for Teachers

Whenever something moves, you can see the change in energy of that system. Energy can make things move or cause a change in the position or state of an object. Energy can be defined as the capacity for doing work. Work is done when a force moves an object over a given distance. The capacity for work, or energy, can come in many different forms. Examples of such forms are mechanical, electrical, chemical or nuclear energy.

This lesson introduces mechanical energy, the form of energy that is easiest to observe on a daily basis. All moving objects have mechanical energy. There are two types of mechanical energy: potential energy and kinetic energy. Potential energy is the energy that an object has because of its position and is measured in Joules (J). Potential energy can also be thought of as stored energy. Kinetic energy is the energy an object has because of its motion and is also measured in Joules (J). Due to the principle of conservation of energy, energy can change its form (potential, kinetic, heat/thermal, electrical, light, sound, etc.) but it is never created or destroyed.

Within the context of mechanical energy, potential energy is a result of an object's position, mass and the acceleration of gravity. A book resting on the edge of a table has potential energy; if you were to nudge it off the edge, the book would fall. It is sometimes called gravitational potential energy (PE). It can be expressed mathematically as follows:

PE = mass x g x height or PE = weight x height

where PE is the potential energy, and g is the acceleration due to gravity. At sea level, g = 9.81 meters/sec2 or 32.2 feet/sec2. In the metric system, we would commonly use mass in kilograms or grams with the first equation. With English units it is common to use weight in pounds with the second equation.

Kinetic energy (KE) is energy of motion. Any object that is moving has kinetic energy. An example is a baseball that has been thrown. The kinetic energy depends on both mass and velocity and can be expressed mathematically as follows:

Here KE stands for kinetic energy. Note that a change in the velocity will have a much greater effect on the amount of kinetic energy because that term is squared. The total amount of mechanical energy in a system is the sum of both potential and kinetic energy, also measured in Joules (J).

Total Mechanical Energy = Potential Energy + Kinetic Energy

Engineers must understand both potential and kinetic energy. A simple example would be the design of a roller coaster — a project that involves both mechanical and civil engineers. At the beginning of the roller coaster, the cars must have enough potential energy to power them for the rest of the ride. This can be done by raising the cars to a great height. Then, the increased potential energy of the cars is converted into enough kinetic energy to keep them in motion for the length of the track. This is why roller coaters usually start with a big hill. As the cars start down the first hill, potential energy is changed into kinetic energy and the cars pick up speed. Engineers design the roller coaster to have enough energy to complete the course and to overcome the energy-draining effect of friction.

Vocabulary/Definitions

conservation of energy: A principle stating that the total energy of an isolated system remains constant regardless of changes within the system. Energy can neither be created nor destroyed.

energy: Energy is the capacity to do work.

kinetic energy: The energy of motion.

mechanical energy: Energy that is composed of both potential energy and kinetic energy.

potential energy: The energy of position, or stored energy.

Associated Activities

Lesson Closure

Restate that both potential energy and kinetic energy are forms of mechanical energy. Potential energy is the energy of position and kinetic energy is the energy of motion. A ball that you hold in your hand has potential energy, while a ball that you throw has kinetic energy. These two forms of energy can be transformed back and forth. When you drop a ball, you demonstrate an example of potential energy changing into kinetic energy.

Explain that energy is an important engineering concept. Engineers need to understand the many different forms of energy so that they can design useful products. An electric pencil sharpener serves to illustrate the point. First, the designer needs to know the amount of kinetic energy the spinning blades need in order to successfully shave off the end of the pencil. Then, the designer must choose an appropriately-powered motor to supply the necessary energy. Finally, the designer must know the electrical energy requirements of the motor in order for the motor to properly do its assigned task.

Assessment

Pre-Lesson Assessment

Discussion Questions: Solicit, integrate and summarize student responses.

  • What are examples of dangerous unsafe placement of objects? (Possible answers: Boulders on the edge of a cliff, dishes barely on shelves, etc.).

Post-Introduction Assessment

Question/Answer: Ask the students and discuss as a class:

  • What has more potential energy: a boulder on the ground or a feather 10 feet in the air? (Answer: The feather because the boulder is on the ground and has zero potential energy. However, if the boulder was 1 mm off the ground, it would probably have more potential energy.)

Lesson Summary Assessment

Group Brainstorm: Give groups of students each a ball (example, tennis ball). Remind them that energy can be converted from potential to kinetic and vice versa. Write a question on the board and have them brainstorm the answer in their groups. Have the students record their answers in their journals or on a sheet of paper and hand it in. Discuss the student groups' answers with the class.

  • How can you throw a ball and have its energy change from kinetic to potential and back to kinetic without touching the ball once it relases from your hand? (Answer: Throw it straight up in the air.)

Calculating: Have students practice problems solving for potential energy and kinetic energy:

  • If a mass that weighs 8 kg is held at a height of 10 m, what is its potential energy? (Answer: PE = (8 kg)*(9.8 m/s2)*(10 m) = 784 kg*m2/s2 = 784 J)
  • Now consider an object with a kinetic energy of 800 J and a mass of 12 kg. What is its velocity? (Answer: v = sqrt(2*KE/m) = sqrt((2 * 800 J)/12 kg) = 11.55 m/s)

Lesson Extension Activities

There is another form of potential energy, not related to height, which is called spring potential or elastic potential energy. In this case, energy is stored when you compress or elongate a spring. Have the students search the Internet or library for the equation of spring potential energy and explain what the variables in the equation represent. The answer is

PEspring = ½ k∙x2

where k is the spring constant measured in N/m (Newton/meters) and x is how far the spring is compressed or stretched measured in m (meters).

References

Argonne Transportation - Laser Glazing of Rails. September 29, 2003. Argonne National Laboratory, Transportation Technology R&D Center. October 15, 2003. http://www.anl.gov/index.html

Asimov, Isaac. The History of Physics. New York: Walker & Co., 1984.

Jones, Edwin R. and Richard L. Childers. Contemporary College Physics. Reading, MA: Addison-Wesley Publishing Co., 1993.

Kahan, Peter. Science Explorer: Motion, Forces, and Energy. Upper Saddle River, NJ: Prentice Hall, 2000.

Luehmann, April. Give Me Energy. June 12, 2003. Science and Mathematics Initiative for Learning Enhancement, Illinois Institute of Technology. October 15, 2003. http://www.iit.edu/~smile/ph9407.html

Nave, C.R. HyperPhysics. 2000. Department of Physics and Astronomy, Georgia State University. October 15, 2003. hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

The Atoms Family - The Mummy's Tomb – Raceways. Miami Museum of Science and Space Transit Planetarium. October 15, 2003. http://www.miamisci.org/af/sln/mummy/raceways.html

Contributors

Bailey Jones; Matt Lundberg; Chris Yakacki; Malinda Schaefer Zarske; Denise Carlson

Copyright

© 2004 by Regents of the University of Colorado.

Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

Acknowledgements

The contents of this digital library curriculum were developed under a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and National Science Foundation GK-12 grant no. 0338326. However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: November 30, 2017

Summary

In this lesson, students are introduced to both potential energy and kinetic energy as forms of mechanical energy. A hands-on activity demonstrates how potential energy can change into kinetic energy by swinging a pendulum, illustrating the concept of conservation of energy. Students calculate the potential energy of the pendulum and predict how fast it will travel knowing that the potential energy will convert into kinetic energy. They verify their predictions by measuring the speed of the pendulum. This engineering curriculum meets Next Generation Science Standards (NGSS).

Engineering Connection

Mechanical engineers are concerned about the mechanics of energy — how it is generated, stored and moved. Product design engineers apply the principles of potential and kinetic energy when they design consumer products. For example, a pencil sharpener employs mechanical energy and electrical energy. When designing a roller coaster, mechanical and civil engineers ensure that there is sufficient potential energy (which is converted to kinetic energy) to move the cars through the entire roller coaster ride.

Ch 7, Energy; Ex 4, 7, 13, 14, 44, 46; Pb 2, 6

Ex7.4 When a rifle with a long barrel is fired, the force of expanding gases acts on the bullet for a longer distance. What effect does this have on the velocity of the emerging bullet? (Do you see why long-range cannons have such long barrels?)

Work = force x distance

work = change in KE = change in [ 1/2 m v2 ]

The expanding gases exert a force on the bullet. This force is exerted over a distance so work is done on the bullet. This work done on the bullet means its KE (kinetic energy) increases. If the distance over which the force acts increases, then the amount of work done on the bullet increases and the final value of its KE increases -- more KE means more speed! Long-rang cannons on naval ships are 10 or 15 meters long.

 

Ex 7.7 Can something have energy without having momentum? Explain. Can something have momentum without having energy? Defend your answer.

An object at rest has no momentum; its velocity is zero. But it can still have potential energy -- PE -- because of work we have done on it. We can lift an object and give it gravitational PE = m g h. It has PE because it can do work on something else as it falls -- or it can increase its own KE. Lifting the object, to give it PE, requires that work be done on it. If we compress a spring, we must do work on it. That work is then stored in the compressed spring and is available to do work on something else. That means there is PE stored in a compressed spring. If we bend a bow we must do work on it. That work is then stored in the bent bow and is available to do work on something else. That means there is PE stored in a bent bow. In all these cases, on object can have energy -- potential energy -- even though it is at rest; an object at rest has no momentum.

However, if an object has momentum it has a velocity. And, if it has a velocity, it has KE -- kinetic energy or energy of motion. So, if an object has momentum, it must also have energy.

 

Ex 7.13 At what point in its motion is the KE of a pendulum bob a maximum? At what point is its PE a maximum? When its KE is half its maximum value, how much PE does it have?

The KE of a pendulum is maximum at the bottom of its swing where its PE is minimum (or where we often take its PE to be zero).

The PE of a pendulum is maximum at the top of its swing where it momentarily stops and its KE is zero.

The total energy remains constant,

So when the pendulum's KE is half its maximum value, its PE is half its maximum value if we have counted its PE as zero at the bottom of its swing.

 

Ex 7.14 A Physics instructor demonstrates energy conservation by releasing a heavy pendulum bob, as shown in the sketch, allowing it to swing to and fro. What would happen if, in his exuberance, he gave the bob a slight shove as it left his nose? Why?

Ouch!

If the bob starts off from rest, it will convert its PE into KE going down to the bottom of the swing and then convert KE back to PE and come to rest at the other end of its swing and then start back. On the way back, it increases its KE by reducing its PE until it comes to the bottom of the swing. On the last part of its swing, this KE decreases as it rises and increases its PE until the KE finally goes to zero and the pendulum comes to rest. With KE = 0, the PE must be equal to the total energy which is the original PE and it is back at its original height.

However, if the bob starts off with some KE because of a shove, then its total energy is greater than for the case just described. At the bottom of its swing, the pendulum will have a greater speed due to this increased energy. At the other end of the swing, it will come to rest at a greater height so its PE is greater. On the way back, that great height means it will smash our exuberant physicist in the nose!

Ouch!

 

Ex 7.44 Scissors for cutting paper have long blades and short handles, whereas metal-cutting shears have long handles and shout blades. Bolt cutters have very long handles and very short blades. Why is this so?

Scissors are levers. A longer blade with the short handles means that the longer blade moves a greater distance but with less force when compared to the handles. Metal-cutting shears have blades that move a smallerdistance but with more force. The bolt cutters, with very short blades and very long handles, have blades that move and even smaller distance but with even greater force. Input work is done by the "operator" as she or he exerts a force on the handles and moves them through a distance. Output work is done by the blade as it exerts a force and moves through a distance. These devices trade off moving a small force through a large distance, as with the scissors, or moving a greater force through a smaller distance.

 

Ex 7.44 Consider the swinging-balls apparatus, also known as Newton's pendulum. If two balls are lifted and released, momentum is conserved as two balls pop out the other side with the same speed as the released balls at impact. But momentum would also be conserved if one ball popped out at twice the speed. Can you explain why this never happens? (And why is this exercise in Chapter 6, on Energy, rather than Chapter 5, on Momentum?)

Indeed, momentumwould be conserved if a single ball popped out with twice the original, incoming speed. We might write that as

But what does that do to the KE? Remember that KE = (1/2) m v2

Because of the way KE is defined, this situation would make the final KE twice as much as the initial KE. So this doe not happen!


Pb 7.2 This question is typical on some driver's license exams: A car moving at 50 km/h skids 15 m with locked brakes. How far will the car skid with locked brakes at 150 km/h?

 

Work done by the brakes changes the car's KE from its initial value to zero. This work is equal to the force of friction causing the skid marks multiplied by the length of the skid marks since

When the car's speed is increased from 50 km/h to 150 km/h, its KE is increased by nine times! Remember that KE is given by

so increasing v by three means that KE increases by nine. The force of friction between the road and the skidding tires does not change appreciably so the distance or length over which the force is applied must increase nine times as well. That means we would expect skid marks to be about 9 x 15 m or 135 m long.

 

Pb 7.6 Consider the inelastic collision between the two freight cars in the last chapter (Figure 6.11). The momentum before and after the collision is the same. The KE, however, is less after the collision than before the collision. How much less, and what becomes of this energy?

The two freight cars have the same mass -- just call it M -- so the momentum is conserved. We can see that from

But what has happened to the KE?

The final Kinetic Energy is only one-half the initial Kinetic Energy (even though the final momentum is equal to the initial momentum). In an accidental collision, kinetic energy will be "lost" (that is, converted into heat) by bending and breaking parts of the cars. In a designed collision, like this, kinetic energy will be "lost" (or turned into heat) by heating up the couplings that hold the cars together.

Here's an "extra" one.

Pb 7.* Your monthly electric bill is probably expressed in kilowatt-hours (kWh), a unit of energy delivered by the flow of 1 kW of electricity for 1 hr. How many joules of energy do you get when you buy 1 kWh?

1 watt = 1 W = 1 J/s

Therefore,

1 J = (1 W) x (1 s) = (1 W) (1 s) = 1 W s

1 kWh = [1 kW] [ h ]

1 kWh = [1 kW (1000 W /kW)] [ h (60 min/h) (60 s/min)] = 3,600,000 Ws = 3,600,000 J

1 kWh = 3,600,000 J

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Typical or possible multiple-guess questions for this material:

1. If you push an object twice as far while applying the same force you do

A) half as much work

B) the same amount of work

C) twice as much work

D) four times as much work

 

2. If you push an object just as far while applying twice the force you do

A) half as much work

B) the same amount of work

C) twice as much work

D) four times as much work

3. Exert 1 N for a distance of 1 m in 1 s and you deliver a power of

A) 0.5 W

B) 1.0 W

C) 2.0 W

D) 3.0 W

 

4. Exert 100 J in 50 s and your power output is

A) 0.5 W

B) 1.0 W

C) 2.0 W

D) 4.0 W

 

5. An object is raised above the ground gaining a certain amount of potential energy. If the same object is raised twice as high it gains

A) half as much energy

B) the same amount of energy

C) twice as much energy

D) four times as much energy

 

6. An object that has kinetic energy must be

A) elevated

B) falling

C) moving

D) at rest

 

7. An object that has potential energy may have this energy because of its

A) speed

B) acceleration

C) momentum

D) position

 

8. A clerk can lift containers a vertical distance of 1 meter or can roll them up a 2 meter-long ramp to the same elevation. With the ramp, the applied force required is about

A) one-fourth as much

B) half as much

C) the same

D) twice as much

 

9. When a car is braked to a stop, its kinetic energy is transformed to

A) energy of motion

B) heat energy

C) stopping energy

D) potential energy

 

10. For which position above does the ball on the end of the string have ?

 

11. For which position above does the ball on the end of the string have ?

 

12. Which requires more work: lifting a 5 kg sack vertically 2 meters or lifting a 10 kg sack vertically 4 meters?

A) lifting the 5 kg sack

B) both require the same amount of work

C) lifting the 10 kg sack

D) both require the same amount of force

 

13. A 10 kg sack is lifted 2 meters in the same time as a 5 kg sack is lifted 4 meters. The power expended in raising the 10 kg sack compared to the power used to lift the 5 kg sack is

A) half as much

B) the same

C) twice as much

D) four times as much

 

14. A 2 kg mass is held 4 m above the ground. What is the approximate potential energy of the mass with respect to the ground?

A) 8 J

B) 40 J

C) 80 J

D) 160 J

 

15. A 2 kg mass has 40 J of potential energy with respect to the ground. Approximately how far is it located above the ground?

A) 1 m

B) 2 m

C) 3 m

D) 4 m

 

16. Using 1,000 J of work, a model elevator is raised from the ground floor to the second floor in 20 seconds. How much power does the elevator use?

A) 50 W

B) 500 W

C) 2 kW

D) 20 kW

 

17. A car moves 4 times as fast as another identical car. Compared to the slower car, the faster car has

A) the same kinetic energy

B) 4 times the kinetic energy

C) 8 times the kinetic energy

D) 16 times the kinetic energy

 

18. A car moving at 50 km/hr skids 20 m with locked brakes. How far will the car skid with locked brakes if it is traveling at 150 km/hr?

A) 40 m

B) 60 m

C) 90 m

D) 180 m

 

19. When a rifle is fired it recoils so both the bullet and rifle are set in motion. The rifle and bullet ideally acquire equal but opposite amounts of

A) kinetic energy

B) momentum

C) potential energy

D) all of the above

 

20. What does an object have when moving that it doesn`t have when at rest?

A) momentum

C) mass

D) all of the above

 

21. If an object has kinetic energy, then it also must have

A) momentum

B) velocity

C) speed

D) all of the above

| Back to 3050's Home Page | Back to Calendar | ToC, Ch 7 | Ch 8, Rotation |

Answers to the typical or possible multiple-choice questions for this material:

1. If you push an object twice as far while applying the same force you do

A) half as much work

B) the same amount of work

D) four times as much work

 

2. If you push an object just as far while applying twice the force you do

A) half as much work

B) the same amount of work

D) four times as much work

 

3. Exert 1 N for a distance of 1 m in 1 s and you deliver a power of

A) 0.5 W

C) 2.0 W

D) 3.0 W

 

4. Exert 100 J in 50 s and your power output is

A) 0.5 W

B) 1.0 W

D) 4.0 W

 

5. An object is raised above the ground gaining a certain amount of potential energy. If the same object is raised twice as high it gains

A) half as much energy

B) the same amount of energy

D) four times as much energy

 

6. An object that has kinetic energy must be

A) elevated

B) falling

D) at rest

 

7. An object that has potential energy may have this energy because of its

A) speed

B) acceleration

C) momentum

 

8. A clerk can lift containers a vertical distance of 1 meter or can roll them up a 2 meter-long ramp to the same elevation. With the ramp, the applied force required is about

A) one-fourth as much

C) the same

D) twice as much

 

9. When a car is braked to a stop, its kinetic energy is transformed to

A) energy of motion

C) stopping energy

D) potential energy

 

10. For which position above does the ball on the end of the string have ?

 

11. For which position above does the ball on the end of the string have ?

 

12. Which requires more work: lifting a 5 kg sack vertically 2 meters or lifting a 10 kg sack vertically 4 meters?

A) lifting the 5 kg sack

B) both require the same amount of work

D) both require the same amount of force

 

13. A 10 kg sack is lifted 2 meters in the same time as a 5 kg sack is lifted 4 meters. The power expended in raising the 10 kg sack compared to the power used to lift the 5 kg sack is

A) half as much

C) twice as much

D) four times as much

 

14. A 2 kg mass is held 4 m above the ground. What is the approximate potential energy of the mass with respect to the ground?

A) 8 J

B) 40 J

D) 160 J

 

15. A 2 kg mass has 40 J of potential energy with respect to the ground. Approximately how far is it located above the ground?

A) 1 m

C) 3 m

D) 4 m

 

16. Using 1,000 J of work, a model elevator is raised from the ground floor to the second floor in 20 seconds. How much power does the elevator use?

B) 500 W

C) 2 kW

D) 20 kW

 

17. A car moves 4 times as fast as another identical car. Compared to the slower car, the faster car has

A) the same kinetic energy

B) 4 times the kinetic energy

C) 8 times the kinetic energy

 

18. A car moving at 50 km/hr skids 20 m with locked brakes. How far will the car skid with locked brakes if it is traveling at 150 km/hr?

A) 40 m

B) 60 m

C) 90 m

 

19. When a rifle is fired it recoils so both the bullet and rifle are set in motion. The rifle and bullet ideally acquire equal but opposite amounts of

A) kinetic energy

C) potential energy

D) all of the above

 

20. What does an object have when moving that it doesn`t have when at rest?

B) energy

C) mass

D) all of the above

 

21. If an object has kinetic energy, then it also must have

A) momentum

B) velocity

C) speed

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